Aristotle's Logic and Metaphysics Roger Bishop Jones Abstract Formalisation in higher order logic of parts of Aristotle's logic and metaphysics. Created 2009/05/21 Last Change Date: 2012/01/23 21:40:02 Id: t028.doc,v 1.33 2012/01/23 21:40:02 rbj Exp http://www.rbjones.com/rbjpub/pp/doc/t028.pdf c Roger Bishop Jones; Licenced under Gnu LGPL Contents 1 Prelude 3 2 Introduction 4 2.1 Preliminary Formalities...................................5 3 Metaphysics (I) 5 3.1 The Grice/Code/Speranza Formulae............................6 3.2 References to Plato......................................6 3.3 Aristotelian References....................................6 3.4 Formal Principles.......................................7 3.4.1 Categories.......................................7 3.4.2 Predication......................................9 3.4.3 The Principles in HOL................................ 10 3.5 Total Definitions....................................... 11 3.6 Partial Definitions...................................... 13 3.7 Ontological Theorems.................................... 14 3.8 Platonic Principles and Theorems.............................. 15 3.9 Some Comments on The Conjectures............................ 16 4 The Organon 18 4.1 Models and Their Significance................................ 18 4.2 Preliminaries......................................... 21 4.2.1 Generating the Syllogisms.............................. 21 4.2.2 The Square of Opposition.............................. 25 4.2.3 Are The Syllogisms Tautologous?.......................... 26 4.3 Interpretation in Set Theory................................. 28 4.3.1 Generating The Propositions............................ 29 4.3.2 Proving the Syllogisms................................ 30 4.4 Propositional Interpretation................................. 31 4.5 Naive Interpretation in Predicate Calculus......................... 33 4.5.1 Semantics....................................... 33 4.5.2 Predication...................................... 33 4.5.3 The Laws of Immediate Inference.......................... 34 4.5.4 The Square of Opposition.............................. 35 4.5.5 The Syllogisms.................................... 35 4.5.6 Generating Syllogisms................................ 36 4.6 Predicate Calculus Without Empty Terms......................... 36 4.6.1 Semantics....................................... 37 4.6.2 Predication...................................... 37 4.6.3 Laws of Immediate Inference............................ 38 4.6.4 The Valid Syllogisms................................. 40 4.6.5 Proving the Syllogisms................................ 40 4.7 Existential Import in Universals............................... 40 4.7.1 Semantics....................................... 40 4.7.2 Predication...................................... 41 4.7.3 The Laws of Immediate Inference.......................... 42 4.7.4 The Square of Opposition.............................. 43 4.7.5 The Syllogisms.................................... 44 4.8 Existential Import in Affirmations............................. 44 4.8.1 Semantics....................................... 44 2 4.8.2 Predication...................................... 45 4.8.3 The Laws of Immediate Inference.......................... 46 4.8.4 The Square of Opposition.............................. 47 4.8.5 The Syllogisms.................................... 47 4.9 Modal Syllogisms....................................... 48 4.9.1 Semantics....................................... 48 4.9.2 Predication...................................... 49 4.9.3 Laws of Immediate Inference............................ 51 4.9.4 The Valid Modal Syllogisms............................. 52 4.9.5 General Results.................................... 53 4.9.6 Proving the Syllogisms................................ 54 4.10 Demonstrative Truth..................................... 55 5 Metaphysics (II) 56 5.1 Semantics........................................... 56 5.2 Predication.......................................... 59 5.3 Propositional Operators................................... 61 5.4 Quantification......................................... 63 5.5 Judgements.......................................... 63 5.6 Conversions.......................................... 64 5.7 Modal Conversions...................................... 65 5.8 Other Conversions...................................... 66 5.9 Syllogisms for Essential Predication............................. 67 5.10 Some Accidental Syllogisms................................. 67 5.11 Grice and Code........................................ 67 5.11.1 Common Material.................................. 67 6 Conclusions 69 7 Postscript 69 A Theory Listings 71 A.1 The Theory ariscat..................................... 72 A.2 The Theory syllog1..................................... 75 A.3 The Theory syllog2..................................... 76 A.4 The Theory syllog3..................................... 77 A.5 The Theory syllog4..................................... 79 A.6 The Theory syllog5..................................... 81 A.7 The Theory syllog6..................................... 83 A.8 The Theory modsyllog................................... 85 A.9 The Theory syllmetap.................................... 87 A.10 The Theory gccon...................................... 91 Bibliography 93 Index 95 1 Prelude This document is intended ultimately to form a chapter of Analyses of Analysis [6]. Some of the material is not expected to be in that history including: 3 • the material up to and including the Prelude • the Postscript and any material following it. • possibly some other parts which have been marked for exclusion My original purpose in preparing this document was to analyse certain semi-formal statements, relating to the philosophy of Aristotle, which were posted to the hist-analytic mailing list (see message in archive) originating primarily in joint work by Grice [4] and Code [3]. This has now been overtaken by various other philosophical motivations. Of these the most important for me at present lie in the perceived relevance of Aristotle's metaphysics to what I am trying elsewhere to write about Metaphysical Positivism. One tentative idea in this exposition involves three comparisons intended to illuminate the tension between essentialism and nominalism and inform the search for a middle ground. These three are between Plato and Aristotle, between Hume and Kant, and between Carnap and one or more twentieth century metaphysicians. For this purpose I seek some kind of understanding of Arstotle's essentialism, and it is for me natural to use formal modelling as one way of realising that understanding. Since my own backround in formal modelling comes from Computer Science and Information Systems Engineering, my own preferred languages, methods and tools, which I believe can be effectively applied to some kinds of philosophical problems, are probably alien to most if not all philosophers, and it is therefore a secondary purpose of this material to try to make this kind of modelling intelligible to some philosophers. This is not a presentation of established methods with proven philosophical benefits. It is an exploration and adaptation of methods from other domains to philosophy, and the benefits, are to be discovered, not merely displayed. The present state of the document is rather rough and ready. Formal modelling takes time, but presenting such material takes longer, and while I am hot on the trail of better, more illuminating models, the presentation will not be polished and transparent. Further discussion of what might become of this document in the future may be found in my postscript (Section7). In this document, phrases in coloured text are hyperlinks, like on a web page, which will usually get you to another part of this document (the blue parts, the contents list, page numbers in the Index) but sometimes take you (the red bits) somewhere altogether different (if you happen to be online) like the hist-analytic archives. For description of the formal languages, methods and tools used in or in producing this document see: [5]. 2 Introduction My purpose here is to use formal models to aid in understanding the philosophies of Plato and Aristotle, both in relation to their contribution generally to the areas of interest, philosophical logic, semantics and metaphysics, and also more specifically in relation to the extent to which these philosophers laid the ground for the distinction which was later expressed in Hume's fork. In doing this I began with some enquiries into Aristotle's metaphysics published by Code [3] and produced from this a preliminary model (Section3). In these an important defect is that the model does not support the u-p syllogisms on which Code's analysis depends more heavily than one might have expected, and also does not allow for modal operators, which not only enter into Code's material 4 but are also important for the kinds of comparison with later philosophers which I had hoped to undertake. I have also failed at this stage to bring out the distinction between Plato and Aristotle. Perhaps more important is that I did not arrive at a good understanding of Code and the model is therefore unlikely to fully reflect his intensions. It is also the case that the method I adopted for the analysis of Code is one which he would have been likely to question. His paper does briefly discuss unfavourably the interpretation of Aristotle in terms of modern idom such as that of set theory. It is